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  • PY3P05

    o  Rotational transitions

    o  Vibrational transitions

    o  Electronic transitions


    o  Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated.

    o  This leads to molecular wavefunctions that are given in terms of the electron positions (ri) and the nuclear positions (Rj):

    o  Involves the following assumptions:

    o  Electronic wavefunction depends on nuclear positions but not upon their velocities, i.e., the nuclear motion is so much slower than electron motion that they can be considered to be fixed.

    o  The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fast- moving electrons.


    "molecule (ˆ r i, ˆ R j ) ="electrons( ˆ r i, ˆ R j )"nuclei( ˆ R j )

  • PY3P05

    o  Electronic transitions: UV-visible

    o  Vibrational transitions: IR

    o  Rotational transitions: Radio

    Electronic Vibrational Rotational



    o  Must first consider molecular moment of inertia:

    o  At right, there are three identical atoms bonded to “B” atom and three different atoms attached to “C”.

    o  Generally specified about three axes: Ia, Ib, Ic.

    o  For linear molecules, the moment of inertia about the internuclear axis is zero.

    o  See Physical Chemistry by Atkins.


    I = miri 2

    i "

  • PY3P05

    o  Rotation of molecules are considered to be rigid rotors.

    o  Rigid rotors can be classified into four types:

    o  Spherical rotors: have equal moments of inertia (e.g., CH4, SF6).

    o  Symmetric rotors: have two equal moments of inertial (e.g., NH3).

    o  Linear rotors: have one moment of inertia equal to zero (e.g., CO2, HCl).

    o  Asymmetric rotors: have three different moments of inertia (e.g., H2O).


    o  The classical expression for the energy of a rotating body is:

    where !a is the angular velocity in radians/sec.

    o  For rotation about three axes:

    o  In terms of angular momentum (J = I!):

    o  We know from QM that AM is quantized:

    o  Therefore, , J = 0, 1, 2, …


    Ea =1/2Ia"a 2


    E =1/2Ia"a 2 +1/2Ib"b

    2 +1/2Ic"c 2


    E = Ja 2

    2Ia + Jb 2

    2Ib + Jc 2



    J = J(J +1)!2


    EJ = J(J +1)! 2I

    , J = 0, 1, 2, …

  • PY3P05

    o  Last equation gives a ladder of energy levels.

    o  Normally expressed in terms of the rotational constant, which is defined by:

    o  Therefore, in terms of a rotational term:


    o  The separation between adjacent levels is therefore

    F(J) - F(J-1) = 2BJ

    o  As B decreases with increasing I =>large molecules have closely spaced energy levels.


    hcB = ! 2

    2I => B = !



    F(J) = BJ(J +1)


    o  Transitions are only allowed according to selection rule for angular momentum:

    "J = ±1

    o  Figure at right shows rotational energy levels transitions and the resulting spectrum for a linear rotor.

    o  Note, the intensity of each line reflects the populations of the initial level in each case.

  • PY3P05

    o  Consider simple case of a vibrating diatomic molecule, where restoring force is proportional to displacement

    (F = -kx). Potential energy is therefore

    V = 1/2 kx2

    o  Can write the corresponding Schrodinger equation as


    o  The SE results in allowed energies



    2µ d2" dx 2

    + [E #V ]" = 0


    2µ d2" dx 2

    + [E #1/2kx 2]" = 0


    µ = m1m2 m1 + m2


    Ev = (v +1/2)!"


    " = k µ


    $ % &

    ' (

    1/ 2

    v = 0, 1, 2, …


    o  The vibrational terms of a molecule can therefore be given by

    o  Note, the force constant is a measure of the curvature of the potential energy close to the equilibrium extension of the bond.

    o  A strongly confining well (one with steep sides, a stiff bond) corresponds to high values of k.


    G(v) = (v +1/2) ˜ v


    ˜ v = 1 2"c

    k µ


    $ % &

    ' (

    1/ 2

  • PY3P05

    o  The lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations.

    o  Transition occur for "v = ±1

    o  This potential does not apply to energies close to dissociation energy.

    o  In fact, parabolic potential does not allow molecular dissociation.

    o  Therefore more consider anharmonic oscillator.


    o  A molecular potential energy curve can be approximated by a parabola near the bottom of the well. The parabolic potential leads to harmonic oscillations.

    o  At high excitation energies the parabolic approximation is poor (the true potential is less confining), and does not apply near the dissociation limit.

    o  Must therefore use a asymmetric potential. E.g., The Morse potential:

    where De is the depth of the potential minimum and


    V = hcDe 1" e "a(R"Re )( )



    a = µ" 2



    $ %


    ' (

    1/ 2

  • PY3P05

    o  The Schrödinger equation can be solved for the Morse potential, giving permitted energy levels:

    where xe is the anharmonicity constant:

    o  The second term in the expression for G increases with v => levels converge at high quantum numbers.

    o  The number of vibrational levels for a Morse oscillator is finite:

    v = 0, 1, 2, …, vmax


    G(v) = (v +1/2) ˜ v " ( ˜ v +1/2)2 xe ˜ v


    xe = a2! 2µ"


    o  Molecules vibrate and rotate at the same time => S(v,J) = G(v) + F(J)

    o  Selection rules obtained by combining rotational selection rule !J = ±1 with vibrational rule !v = ±1.

    o  When vibrational transitions of the form v + 1 ! v occurs, !J = ±1.

    o  Transitions with !J = -1 are called the P branch:

    o  Transitions with !J = +1 are called the R branch:

    o  Q branch are all transitions with !J = 0


    S(v,J) = (v +1/2) ˜ v + BJ(J +1)


    ˜ v P (J) = S(v +1,J "1) " S(v,J) = ˜ v " 2BJ


    ˜ v R (J) = S(v +1,J +1) " S(v,J) = ˜ v + 2B(J +1)

  • PY3P05

    o  Molecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 – 4000cm-1 0.01 to 0.5 eV).

    o  Vibrational transitions accompanied by rotational transitions. Transition must produce a changing electric dipole moment (IR spectroscopy).

    P branch

    Q branch

    R branch


    o  Electronic transitions occur between molecular orbitals.

    o  Must adhere to angular momentum selection rules.

    o  Molecular orbitals are labeled, ", #, $, … (analogous to S, P, D, … for atoms)

    o  For atoms, L = 0 => S, L = 1 => P o  For molecules, % = 0 => ", % = 1 => #

    o  Selection rules are thus

    $% = 0, ±1, $S = 0, $"=0, $& = 0, ±1

    o  Where & = % + " is the total angular momentum (orbit and spin).


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